A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 18, 15, 1, 0, 1, 58, 129, 37, 1, 0, 1, 179, 877, 646, 83, 1, 0, 1, 543, 5280, 8030, 2685, 177, 1, 0, 1, 1636, 29658, 82610, 56285, 10002, 367, 1, 0, 1, 4916, 159742, 756218, 919615, 335162, 34777, 749, 1, 0

Offset

0,8

Comments

Generalized Eulerian numbers A008292.

Reversal of A211399. - Philippe Deléham, Feb 12 2013

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

T(n, k) = (n-k)*T(n-1, k-1) + (2*k+1)*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0.

Sum_{k>=0} T(n, k)*2^k = A001147(n).

Sum_{k>=0} T(n, k) = A014307(n). - Philippe Deléham, Mar 19 2005

Example

Triangle begins:
  1;
  1,  0;
  1,  1,   0;
  1,  5,   1,  0;
  1, 18,  15,  1, 0;
  1, 58, 129, 37, 1, 0; ...

Mathematica

T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* G. C. Greubel, Jun 30 2017 *)

Crossrefs

Diagonals: A000007, A000012, A050488, A142965, A142966.

Columns: A000012, A000340, A156922, A156923, A156924.

Sequence in context: A211399 A265192 A157012 * A269945 A322013 A102259

Adjacent sequences: A102362 A102363 A102364 * A102366 A102367 A102368

Keyword

nonn,easy,tabl

Author

Philippe Deléham, Feb 22 2005

Status

approved